Insofar as the book of nature is written in the language of
mathematics (as Galileo once said), most of the sentences are partial
differential equations. The purpose of this course is to introduce
you to these important equations - their origins, applications, and
how to solve them. In addition to being ubiquitous, many partial
differential equations (PDEs) are difficult to solve. We will develop
the mathematical theory of PDEs, and systematically explore several
of the most well-known cases. Throughout the course I will attempt to
strike a balance between the mathematical properties of the equations
or their solutions, and the physical implications; in many instances
your physical intuition corresponds to mathematical facts - and vice
versa!

The material we will cover can be divided into three topics:

I. The physical origins of PDEs

II. PDEs on unbounded domains

III. PDEs on bounded domains

A main focus of the course will be The Big Four: the
transport, heat, wave, and Laplace equations. In addition we will
cover expansions in orthogonal functions, and the use of Fourier and
Laplace transforms in solving PDEs. In the beginning we will
essentially follow Logan, and we will use Haberman more as the
semester progresses. The structure of the course will follow Logan's
text.